The growth of valuations on rational function fields in two variables

Given a valuation on the function field $k(x,y)$, we examine the set of images of nonzero elements of the underlying polynomial ring $k[x,y]$ under this valuation. For an arbitrary field $k$, a Noetherian power series is a map $z:\mathbb{Q}\to k$ that has Noetherian (i.e., reverse well-ordered) support. Each Noetherian power series induces a natural valuation on $k(x,y)$. Although the value groups corresponding to such valuations are well-understood, the restrictions of the valuations to underlying polynomial rings have yet to be characterized. Let $\Lambda_n$ denote the images under the valuation $v$ of all nonzero polynomials $f \in k[x,y]$of at most degree $n$ in the variable $y$. We construct a bound for the growth of $\Lambda_n$ with respect to $n$ for arbitrary valuations, and then specialize to valuations that arise from Noetherian power series. We provide a sufficient condition for this bound to be tight.